For a rational number $r\in(0,1)$ the number $z=\sin(r\pi)$ is an algebraic number — such numbers appear to be called trigonometric numbers. What is its degree? That is, what is the minimal degree of a polynomial with integer coefficients whose zero $z$ is?
A few observations are relatively simple for $r=p/q$ with $\gcd(p,q)=1$:
If $q$ is not a product of distinct Fermat primes and a power of two, then $z$ is not constructible and therefore has to have degree at least three.
If $q$ is even, then $P(z)=0$ for the polynomial $P(x)=C_q(1-x^2)-(-1)^p$, where the polynomial $C_q$ is related to the Chebyshev polynomial by $T_q(x)=C_q(x^2)$. The degree of this $P$ is $q$.
If $q$ is odd, then $P(z)=0$ for the polynomial $P(x)=(1-x^2)C_q(1-x^2)^2-1$, where the polynomial $C_q$ is related to the Chebyshev polynomial by $T_q(x)=xC_q(x^2)$. The degree of this $P$ is $2q$.
These observations give some bounds on the degree of the minimal polynomial of $z$. Is something better known? Do we know, for example, that the degree of $z$ always divides $q$?
Unfortunately this question is difficult to search for on Google, as the results are dominated by trigonometric functions with radians and degrees. This question is related to trying to understand the lengths of periodic billiards trajectories in the disc, and this is why only the sine function interests me.
There is a closely related question for the cosine but I am specifically interested in the sine here.
